Contact Geometry, Fano Orbifolds, and Einstein Metrics

Boyer和Galicki教授建议研究几何和拓扑学的几个项目。所有项目的目标是研究黎曼几何中的基本问题，其中有两个主要焦点：Fano簇上的二重丛的接触几何和一些特殊（即，爱因斯坦，正里奇曲率）度量的空间。这里提出的问题和问题深深植根于主要研究者的早期工作，利用了接触几何的Sasakian-Einstein空间和两种Kaehler几何，即Q-阶乘Fano品种与Kaehler-Einstein orbifold度量与正标量曲率，和Calabi-Yau流形与他们的Kaehler-Ricci-flat度量之间的基本关系。最近的工作的主要研究人员导致了一个重要的突破，在研究这种结构时，使用最近的结果Demailly和Kollar，主要的propagatorswere能够构建新的例子，紧凑的爱因斯坦流形在5维以及许多积极的爱因斯坦度量家庭的rationalhomology 7-sphere。主要研究人员使用的技术借用从几个不同的领域;代数几何的森理论和intersection理论，分析的卡拉比猜想，最后的经典微分拓扑的联系孤立hypersurfacesingularities。这些方法可以进一步向各个方向推广。在这方面突出的一个例子是关于奇异球上爱因斯坦度规的存在问题，这是本提案的主要目标之一。更一般地说，主要研究人员希望解决几个分类问题，紧凑的Sasakian-Einstein流形在5和7维。这两个方面的重要性有两个不同的原因。鉴于早期的工作更高的维度的例子可以使用连接构造。与此同时，这两个奇怪的维度似乎在超弦理论中扮演着特殊的角色。结合弦理论和M理论的最新发展，主要研究者还提出了研究四维自对偶Einstein度量和7维、8维特殊完整度量的一些相关问题。虽然这一奋进并不是由技术进步直接推动的，但数学的历史充分表明，今天的纯数学往往成为明天的应用数学。事实上，在这个项目中考虑的数学目前正被用于基本粒子物理学领域，更具体地说，在试图理解宇宙基本力的统一描述。

ABSTRACT DMS - 0203219.Professors Boyer and Galicki propose to investigate several projects in geometry and topology. The objective of all the projects is to study fundamental questions in Riemannian Geometry with two main focal points: Contact Geometry oforbifold bundles over Fano varieties and the existence of somespecial (i.e., Einstein, positive Ricci curvature) metrics on such spaces. The questions and problems proposed here are deeply rooted in theprincipal investigators' earlier work which exploited a fundamental relationship between contact geometry of Sasakian-Einstein spaces and two kinds of Kaehler geometry, namely Q-factorial Fano varieties with Kaehler-Einstein orbifold metrics with positive scalar curvature, and Calabi-Yau manifolds with their Kaehler Ricci-flat metrics. The most recent work of the principal investigators has led to an important breakthrough in the study of such structures when,using recent results of Demailly and Kollar, the principal investigatorswere able to construct new examples of compact Einstein manifolds in dimension 5 as well as many positive Einstein metrics on families of rationalhomology 7-spheres. The techniques used by the principal investigators borrow fromseveral different fields; the algebraic geometry of Mori theory and intersectiontheory, the analysis of the Calabi Conjecture, and finally theclassical differential topology of links of isolated hypersurfacesingularities. These methods can be extended much further and invarious directions. An example that stands out in this respect is the question of the existence of Einstein metrics on exotic spheres, which is one of the main objectives of this proposal. More generally the principal investigators want to address several classification problems concerning compact Sasakian-Einstein manifolds in dimensions 5 and 7. These two dimensions are important for two separate reasons. In view of earlier work higher dimensional examples can be constructed using the join construction. At the same time these two odd dimensions appear to play special role in Superstring Theory. In the context of recent developements inString and M-Theory the principal investigators also propose toinvestigate some related problems concerning self-dual Einstein metrics in dimension 4 and exceptional holonomy metrics in dimension 7 and 8.This project is intended to further the understanding of the mathematicsof certain important types of geometry. While this endeavor is notdirectly motivated by advances in technology, the history of mathematicsadequately demonstrates that today's pure mathematics often becomestomorrow's applied mathematics. Indeed the mathematics being considered inthis project is currently being used in the field of Elementary ParticlePhysics, more specifically, in the attempts at understanding a unifieddescription of the fundamental forces of the universe.-